Question

The solution of a certain differential equation is of the form y(t)=aexp(7t)+bexp(8t) where a and b are constants. The solution has initial conditions y(0)=5 and y' (0)=5. Find the solution by using the initial conditions to get linear equations for a and b. y(t)= ?

Answer 1

To find the constants a and b in the differential equation's solution, apply the initial conditions y(0)=5 and y'(0)=5 to get two linear equations, which are then solved simultaneously to determine a and b.

Step-by-step explanation:

The solution to the given differential equation with initial conditions is found by applying these conditions to the general form of the solution y(t)=aexp(7t)+bexp(8t). Using the initial condition y(0)=5, we substitute t=0 into the equation to get the first linear equation for a and b. Similarly, we find the first derivative y'(t) to apply the second initial condition y'(0)=5, which gives us the second linear equation.

Applying y(0)=5, we get:
a + b = 5.

The first derivative of y(t) is y'(t)=7aexp(7t)+8bexp(8t). Applying y'(0)=5, we get:
7a + 8b = 5.

Solving these linear equations simultaneously, we find the values of a and b, which will then be substituted back into the general solution to obtain the specific solution for y(t).